Abstract Algebra

Introduction
Part 1.  A First Course
	Chapter 1. Set theory
		1.1. Sets and functions
		1.2. Relations
		1.3. Binary operations
	Chapter 2. Group theory
		2.1. Groups
		2.2. Subgroups
		2.3. Cyclic groups
		2.4. Generators
		2.5. Direct products
		2.6. Groups of isometries
		2.7. Symmetric groups
		2.8. Homomorphisms
		2.9. The alternating group
		2.10. Cosets
		2.11. Conjugation
		2.12. Normal subgroups
		2.13. Quotient groups
	Chapter 3. Ring theory
		3.1. Rings
		3.2. Families of rings
		3.3. Units
		3.4. Integral domains
		3.5. Ring homomorphisms
		3.6. Subrings generated by elements
		3.7. Fields of fractions
		3.8. Ideals and quotient rings
		3.9. Principal ideals and generators
		3.10. Polynomial rings over fields
		3.11. Maximal and prime ideals
	Chapter 4. Advanced group theory
		4.1. Isomorphism theorems
		4.2. Commutators and simple groups
		4.3. Automorphism groups
		4.4. Free abelian groups
		4.5. Finitely generated abelian groups
		4.6. Group actions on sets
		4.7. Permutation representations
		4.8. Burnside's formula
		4.9. p-groups
		4.10. The Sylow theorems
		4.11. Applications of Sylow theory
		4.12. Simplicity of alternating groups
		4.13. Free groups and presentations
	Chapter 5. Advanced ring theory
		5.1. Unique factorization domains
		5.2. Polynomial rings over UFDs
		5.3. Irreducibility of polynomials
		5.4. Euclidean domains
		5.5. Vector spaces over fields
		5.6. Modules over rings
		5.7. Free modules and generators
		5.8. Matrix representations
	Chapter 6. Field theory and Galois theory
		6.1. Extension fields
		6.2. Finite extensions
		6.3. Composite fields
		6.4. Constructible numbers
		6.5. Finite fields
		6.6. Cyclotomic fields
		6.7. Field embeddings
		6.8. Algebraically closed fields
		6.9. Transcendental extensions
		6.10. Separable extensions
		6.11. Normal extensions
		6.12. Galois extensions
		6.13. Permutations of roots
Part 2.  A Second Course
	Chapter 7. Topics in group theory
		7.1. Semidirect products
		7.2. Composition series
		7.3. Solvable groups
		7.4. Nilpotent groups
		7.5. Groups of order p3
	Chapter 8. Category theory
		8.1. Categories
		8.2. Functors
		8.3. Natural transformations
		8.4. Limits and colimits
		8.5. Adjoint functors
		8.6. Representable functors
		8.7. Equalizers and images
		8.8. Additive and abelian categories
	Chapter 9. Module theory
		9.1. Associative algebras
		9.2. Homomorphism groups
		9.3. Tensor products
		9.4. Exterior powers
		9.5. Graded rings
		9.6. Determinants
		9.7. Torsion and rank
		9.8. Noetherian rings and modules
		9.9. Modules over PIDs
		9.10. Canonical forms
	Chapter 10. Topics in Galois theory
		10.1. Norm and trace
		10.2. Discriminants
		10.3. Extensions by radicals
		10.4. Linearly disjoint extensions
		10.5. Normal bases
		10.6. Profinite groups
		10.7. Infinite Galois theory
	Chapter 11. Commutative algebra
		11.1. Localization
		11.2. Local rings
		11.3. Integral extensions
		11.4. Radicals of ideals
		11.5. Going up and going down
		11.6. Primary decomposition
		11.7. Hilbert's Nullstellensatz
		11.8. Spectra of rings
		11.9. Krull dimension
		11.10. Dedekind domains
		11.11. Discrete valuation rings
		11.12. Ramification of primes
	Chapter 12. Homological algebra
		12.1. Exact sequences
		12.2. The snake and five lemmas
		12.3. Homology and cohomology
		12.4. Projective and injective objects
		12.5. Exact functors
		12.6. Projective and injective resolutions
		12.7. Derived functors
		12.8. Tor and Ext
		12.9. Group cohomology
		12.10. Galois cohomology
	Chapter 13. Representation theory
		13.1. Semisimple modules
		13.2. Representations of groups
		13.3. Maschke's theorem
		13.4. Characters
		13.5. Character tables
		13.6. Induced representations
		13.7. Applications to group theory